Steering Through Contact: A Finite-Support Motion Model for Single-Track Center-Articulated Robots

Mohamed Dhia Ounally; Nicolas Samson; Mathis Turgeon-Roy; Veronica Vannini; Johann Laconte; François Pomerleau

cs.ROSep 20, 2026

Steering Through Contact: A Finite-Support Motion Model for Single-Track Center-Articulated Robots

Authors: Mohamed Dhia Ounally, Nicolas Samson, Mathis Turgeon-Roy, Veronica Vannini, Johann Laconte, François Pomerleau

Organizations: Norlab, Universit´e Laval, Qu´ebec, QC, Canada · TSCF, INRAE, Clermont-Ferrand, France

Abstract

Trajectory planning and control in field robotics rely on predicting how propulsion and steering affect vehicle motion when contact points undergo slip. For articulated vehicles, the point-contact kinematic model (PCK) accounts for the linkage geometry but neglects the rotational resistance distributed along the contacts. We propose a finite-support quadratic model (FSQ) for single-track, center-articulated vehicles that incorporates this resistance through a quasi-static balance of lateral slip. Our approach generalizes the standard PCK formulation by relaxing the contact-point assumption. An exact reduction of the quadratic slip cost to contact moments gives a compact closed-form solution for real-time prediction of lateral velocity and yaw rate. We evaluate the proposed method in real-world experiments across asphalt, grass, ice, and mixed routes, using more than 7 km of data. For five-second predictions, FSQ reduces the weighted median translation and yaw errors by 53.5% and 68.9%, respectively, relative to PCK.

Figures & tables

Fig. 1: Articulated motion prediction during a field turn. The purple point-contact prediction, based on pointwise no-slip constraints, deviates markedly from the black traveled path. Blue annotations show longitudinal body velocity $v_{b}$ and articulation $\varphi$ in the overhead view, while the lower-left inset identifies the two actuation mechanisms: the driven track and hydraulic hitch.

Fig. 2: Articulated geometry and lateral contact slip. The tractor is depicted in teal and the trailer in plum, with body frames

\mathcal{F}_{i}

centered at

P_{i}

. Distances

\ell_{i}

connect these origins to hitch

H

\varphi

is articulation, and

v_{b}

is tractor longitudinal velocity. Headings

\theta_{i}

are measured from the world

x_{W}

direction. Dashed lines perpendicular to

x_{i}

lead to the off-frame ICR ( ICR ), with turning radius

r_{\mathrm{PCK}}

for steady point-contact motion. Gray guides locate the contact domains

\mathcal{P}_{i}

enlarged below, with both inset frames oriented alike. In the insets, curved arrows indicate positive yaw rates

\omega_{i}

, and straight arrows show the contribution of rotation about

P_{i}

to lateral slip

s_{\perp,i}

Fig. 3: Field platform and instrumentation over a lidar map. Green labels identify (A) the RS Airy lidar, (B) the Hesai XT-32 lidar, (C) the Xsens MTi-100 IMU ( IMU ), and (D) the ZED 2i camera. A provides articulation and trailer pose estimates in $\mathrm{SE}(3)$ , while measurements from B , C , and D are combined to estimate tractor odometry. Blue insets locate the driven track and hydraulic hitch, with body velocity $v_{b}$ and articulation angle $\varphi$ .

	Characteristic	Symbol	Value	Unit
Robot	Tractor mass	$m_{1}$	$365$	$\mathrm{kg}$
	Trailer mass	$m_{2}$	$75$	$\mathrm{kg}$
	Front center–hitch distance	$\ell_{1}$	$0.8342$	$\mathrm{m}$
	Hitch–rear center distance	$\ell_{2}$	$1.5145$	$\mathrm{m}$
	Front contact length	$l_{t}$	$1.225$	$\mathrm{m}$
	Rear axle separation	$d_{r}$	$0.455$	$\mathrm{m}$

TABLE I: Platform geometry and contact parameters. Only the rear/front contact-weight ratio is fitted from motion data.

Fig. 4: Prediction error versus horizon. Lines and bands show weighted medians and the IQR ( IQR ) of translation RPE ( RPE ) at

P_{1}

(top) and tractor yaw RPE (bottom), with equal weight per run. Triangles mark bands extending beyond the view.

Fig. 5: Five-second prediction errors and contact calibration. The violins show translation RPE (top) and yaw RPE (bottom) on logarithmic axes, with width representing error density. White marks, boxes, and whiskers indicate weighted medians, IQR , and fifth to 95 th percentiles, with equal total weight per run within each group. Triangles mark individual segment errors outside the view. FSQ uses a global rear/front contact-weight ratio

\alpha

or a separate ratio fitted on each terrain’s evaluated runs.

Fig. 6: Five-second prediction errors grouped by longitudinal velocity. At each plotted velocity, lines and bands summarize the weighted medians and IQR of segments whose starting velocity lies within

\pm 0.75\text{\,}\mathrm{m}\text{\,}{\mathrm{s}}^{-1}

of that value, with equal total weight per run in that neighborhood. Direction and speed may change during prediction. Dashed lines mark zero and

|v_{b}|=

1\text{,}\mathrm{m}\text{,}{\mathrm{s}}^{-1}$$ , while triangles indicate IQR extending beyond the vertical limits.

Fig. 7: Campus trajectory with three five-second prediction examples. Boxes on the mapped reference route locate (a) a near-straight segment, (b) a sharp turn, (c) a milder turn. Background point colors indicate height. Insets use a common metric scale, with dashed black for the reference, purple for the point-contact model ( PCK ), and orange for the FSQ ( FSQ ). Hollow circles mark the common starts and arrowheads the endpoints. Scale bars show $50\text{\,}\mathrm{m}$ above and $5\text{\,}\mathrm{m}$ below.

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May 7, 2026J/K move · Enter open · S save

On the Emergence of Pendular Structure in Multi-Contact Locomotion

Lingxue Lyu, Zihui Liu

School of Engineering and Applied Science, University of Pennsylvania, Philadelphia, PA 19104, USA · Department of Aeronautics & Astronautics, Stanford University, Stanford, CA 94305, USA

Legged Robotics Trajectory Optimization Momentum Methods Optimal Control Quadruped Locomotion

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LIPM is everywhere in legged-locomotion control, but almost always as a modeling choice rather than as something the controller's cost actually prefers. This note tries to make that link more explicit. Working from a small centroidal OCP that penalizes the rate of angular momentum, we look at what its optimum tends to look like. Three things come out. With full-rank stance, the optimum drifts toward a pendular force pattern at a rate determined by the SVD of the moment Jacobian; the constant is set by foot-span geometry and matches the experiments to within 16%. With N=2 stance, as in trot, the friction cone introduces a lower bound on

\|\dot{H}_G\|

that no amount of weight tuning fixes; we also see a non-smooth feasibility kink at a critical horizontal acceleration that we can write in closed form. Adding a task term that asks for a nonzero

\dot{H}_G

moves the optimum off the pendular set in a predictable way. None of this is far from the classical ZMP/DCM picture. We test these claims on a point-mass quadruped and on the Unitree Go1 in MuJoCo (open-loop QP and a torque-level closed-loop controller), and we note where the asymptotic story stops being a good description of what the closed loop actually does.

Abstract

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